
\(\tau _{t}\)  is a scalar slow-moving component satisfying \(E\left( \tau _{t}\right) = 1\) , a common synthesis of low–frequency features of the \(K\)  series, possessing a smooth behavior, similarly to what was done in Section , in the univariate case. Also in this case, when \(\tau _t=1, \ \forall t\) , we are in the base \(\mathsf {vMEM}\)  of [1]}.
The semi–nonparametric vector MEM \(\mathsf {SpvMEM}\)  was derived in [2]}, extending the [3]} approach illustrated in Section  to the multivariate case: here it will be extended in ways to be seen in Section ;

\(\mathbf {\mu }\)  is the \(K \times 1\)  vector of the unconditional means;

\(\mathbf {\xi }_{t}\)  is a \(K \times 1\)  vector of interconnected short-run components satisfying \(E\left(\mathbf {\xi }_{t}\right) = {1}\) .
More specifically, the \(K\) -dimensional counterpart of Equation (REF ) is
\(\mathbf {\xi }_t = \left[ {1} - \left( \mathbf {\beta }_1 + \mathbf {\alpha }_1 + \frac{\mathbf {\gamma }_1}{2} \right) {1} \right] +\mathbf {\beta }_1 \mathbf {\xi }_{t-1} + \mathbf {\alpha }_1 \mathbf {x}_{t-1}^{(\xi )} + \mathbf {\gamma }_1 \mathbf {x}_{t-1}^{(\xi -)},\) 
where
\(x_{t,i}^{(\xi )} = \frac{x_{t,i}}{\mu _{i} \tau _t}\qquad {}x_{t,i}^{(\xi -)} = x_{t,i}^{(\xi )} \mathbb {1}^{-}_{t}\qquad {}i = 1, \ldots , K\) 
and \(\mathbf {\beta }_1\) , \(\mathbf {\alpha }_1\)  and \(\mathbf {\gamma }_1\)  are in general \(K \times K\)  matrices.
In the base formulation, \(\mathbf {\beta }_1\)  and \(\mathbf {\gamma }_1\)  are diagonal, but \(\mathbf {\alpha }_1\)  is allowed to be full in order to capture interdependency structures.
Paralleling (REF ), an equivalent specification is
\(\mathbf {\xi }_t = ({1} - \mathbf {\beta }_1^{*} {1}) + \mathbf {\beta }_1^{*} \mathbf {\xi }_{t-1} + \mathbf {\alpha }_1 \mathbf {v}_{t-1} + \mathbf {\gamma }_1 \mathbf {v}^{(-)}_{t-1},\) 
where
\(v_{t, i} = x^{(\xi )}_{t, i} - \xi _{t, i},\qquad {}v_{t, i}^{(-)} = x^{(\xi -)}_{t, i} - \frac{\xi _{t, i}}{2},\qquad {}i = 1, \ldots , K\) 
and \(\mathbf {\beta }_1^{*} = \mathbf {\beta }_1 + \mathbf {\alpha }_1 + \mathbf {\gamma }_1 / 2\) ;

\(\mathbf {\varepsilon }_{t}\)  is a \(K \times 1 \)  conditionally homoskedastic unit mean error term,
\(\mathbf {\varepsilon }_t \overset{\operatorname{i.i.d.}}{\sim } d^{+}({1}, \mathbf {\Sigma });\) 
the actual form of the multivariate distribution \(d^+\)  is left unspecified, with the proviso that it is not straighforward to find a suitable parametric specification. As a matter of fact, as pointed out by [4]}, one could resort to a copula function–based solution (more details provided below) or a multivariate Log–Normal, later used also in [5]}.

